The product of two two digit natural numbers is 2160 and their highest common factor (HCF) is 12. Using this information, determine which pair of numbers among the options represents the correct two numbers.

Introduction / Context:
This question combines the use of product, highest common factor (HCF), and simple factorization to determine two unknown numbers. Knowing the product and the HCF provides a strong structure for working backwards. Such questions are common in number system and arithmetic sections of competitive exams.

Given Data / Assumptions:

• The two numbers are two digit natural numbers.
• Their product is 2160.
• Their HCF is 12.
• We need to find the correct ordered pair from the given options.

Concept / Approach:
If the HCF of two numbers is 12, we can represent them as 12x and 12y, where x and y are co prime integers (their HCF is 1). The product of the two numbers is then 12 * 12 * x * y = 144xy. Since this product is given as 2160, we can solve for xy and then find co prime factor pairs of that value that produce two digit numbers 12x and 12y.

Step-by-Step Solution:
Let the numbers be 12x and 12y. Given product: 12x * 12y = 2160. So 144xy = 2160. Therefore xy = 2160 / 144 = 15. Now find co prime factor pairs of 15: (1, 15) and (3, 5). Corresponding number pairs: (12 * 1, 12 * 15) = (12, 180) and (12 * 3, 12 * 5) = (36, 60). Only (36, 60) gives two digit numbers for both entries.

Verification / Alternative check:
Check HCF and product: HCF(36, 60) = 12. Product: 36 * 60 = 2160. The pair (36, 60) satisfies both conditions exactly. No other candidate pair among the options matches both the HCF and product constraints with both numbers being two digit.

Why Other Options Are Wrong:
(12, 60) has product 720, not 2160, and 12 is not a two digit number. (72, 30) has product 2160 but HCF is 6, not 12. (60, 72) also has product 4320, which is not equal to 2160, and its HCF is 12 but the product is incorrect. (18, 120) includes a three digit number and its HCF and product do not match the required values.

Common Pitfalls:
Many students forget that the co factors x and y must be co prime when the HCF is already factored out. Another mistake is not enforcing the two digit constraint or miscalculating the product. Always express numbers in the form HCF times co prime factors and use the product to determine these co factors.

Final Answer:
(36, 60)